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Cambridge IGCSE0580

Transformations

Mathematics 0580 Chapter Notes

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Transformations
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In this chapter

  1. 1Translation by a vector
  2. 2Reflection in a mirror line
  3. 3Rotation about a centre
  4. 4Enlargement from a centre
  5. 5Describing a single transformation
  6. 6Negative scale factors
  7. 7Combinations of transformations

1. Translation by a vector

A translation moves every point of a shape the same distance in the same direction. The shape does not turn or change size.

A translation is given as a column vector. The top number is the move in the x-direction. The bottom number is the move in the y-direction.

A positive top number moves right and a negative one moves left. A positive bottom number moves up and a negative one moves down.

beginpmatrix a b endpmatrix: (x, y) → (x + a, y + b)

Examiner insight

“Some had the x and y directions the wrong way round in the vector.” “A good number of candidates were able to correctly draw the required translation although common errors included getting only the horizontal or vertical component correct, with a small number omitting this part.”

Worked example 13 marks

Do not use a calculator. Triangle T has vertices A(1, 2), B(4, 2) and C(1, 5). It is translated by the vector beginpmatrix -3 4 endpmatrix. Write down the coordinates of the vertices of the image.

  1. 1

    every x-coordinate changes by -3 and every y-coordinate by +4 [1]

  2. 2

    A' = (1 - 3, 2 + 4) = (-2, 6) [1]

  3. 3

    B' = (1, 6) and C' = (-2, 9) [1]

2. Reflection in a mirror line

A reflection flips a shape in a mirror line. Each point and its image are the same distance from the mirror line, on opposite sides of it.

To describe a reflection fully, name it and give the equation of the mirror line. A vertical line is x = a number. A horizontal line is y = a number.

On Extended papers the mirror line can be any straight line. The line y = x is the diagonal through (0, 0), (1, 1) and (2, 2).

x = a: (x, y) → (2a - x, y)

y = b: (x, y) → (x, 2b - y)

y = x: (x, y) → (y, x)

Examiner insight

“The line of reflection proved slightly more difficult and was often not given, with other common errors of y-axis, x = 0 and x = –1.” “Many candidates drew the reflection correctly but the mirror line of y = x caused difficulty and confusion for others and errors included reflection in the y-axis or x-axis.”

Worked example 13 marks

Do not use a calculator. The points A(-3, 2) and B(0, 6) are reflected in the line x = 1. Write down the coordinates of the images of A and B.

  1. 1

    A is 1 - (-3) = 4 units to the left of the line [1]

  2. 2

    A' is 4 units to the right: A' = (5, 2) [1]

  3. 3

    B is 1 unit left, so B' = (2, 6) [1]

3. Rotation about a centre

A rotation turns a shape about a fixed point, the centre of rotation, through an angle. The shape stays the same size.

To describe a rotation fully, give three things: the angle with its direction, such as 90° clockwise, and the centre. A turn of 180° needs no direction.

On Core papers the centre is the origin, a vertex of the shape or the midpoint of one of its edges. On Extended papers it can be any point.

90^(°)clockwise aboutO: (x, y) → (y, -x)

90^(°)anticlockwise aboutO: (x, y) → (-y, x)

180^(°)aboutO: (x, y) → (-x, -y)

Examiner insight

“While many scored full credit, the centre of rotation caused quite a few problems and there were many different centres given.” “For example, some candidates gave the centre of rotation as (0, –1) or the angle of rotation as 90° clockwise, rather than anti-clockwise.”

Worked example 13 marks

Do not use a calculator. The point P(4, 2) is rotated through 90° anticlockwise about the origin. Write down the coordinates of the image of P.

  1. 1

    P is to the right and above the origin. A quarter turn anticlockwise moves it to the upper left [1]

  2. 2

    (x,y) → (-y,x) [1]

  3. 3

    P' = (-2, 4) [1]

4. Enlargement from a centre

An enlargement changes the size of a shape and keeps its shape. It has a centre of enlargement and a scale factor.

Every length is multiplied by the scale factor. A scale factor above 1 makes the image larger. A fraction between 0 and 1 makes it smaller, and it is still an enlargement.

Each image point lies on the line from the centre through the object point. Its distance from the centre is the scale factor times the object's distance.

k = (image length)/(object length)

image point = centre + k × (object point - centre)

Examiner insight

“Enlargement was usually given but a few were confused by the fact that the image was smaller than the object shape and gave descriptions such as a reduction.” “The identification of the centre of enlargement proved the more challenging with a significant number omitting this part, and (0, 0), (2, 0) and (–2, 0) being common errors.”

Worked example 13 marks

Do not use a calculator. The point P(6, 5) is enlarged with scale factor 1/2 and centre (2, 1). Write down the coordinates of the image of P.

  1. 1

    vector from the centre to P: (6 - 2, 5 - 1) = (4, 4) [1]

  2. 2

    multiply by the scale factor: 1/2 × (4, 4) = (2, 2) [1]

  3. 3

    add to the centre: P' = (2 + 2, 1 + 2) = (4, 3) [1]

5. Describing a single transformation

Questions ask for the single transformation that maps one shape onto another. Give one transformation only. A second transformation spoils the marks already earned.

Name it first, then give its details. A reflection needs its mirror line. A rotation needs its angle, direction and centre. An enlargement needs its centre and scale factor. A translation needs its column vector.

In the 2026 papers a reflection or a translation earned 2 marks. A rotation or an enlargement earned 3.

Examiner insight

“Weaker responses gave a combined transformation of enlargement and translation and so could not gain any marks.” “A few candidates in all three parts gave additional transformations such as rotation followed by a translation and in these cases any marks earned are then spoiled by this additional incorrect work.”

Worked example 13 marks

Do not use a calculator. Triangle A has vertices (1, 1), (2, 1) and (1, 2). Its image B has vertices (1, 1), (3, 1) and (1, 3). Describe fully the single transformation that maps A onto B.

  1. 1

    B has the same shape but is larger: an enlargement [1]

  2. 2

    scale factor = 2/1 = 2 [1]

  3. 3

    the vertex (1, 1) does not move, so the centre is (1, 1) [1]

6. Negative scale factors

With a negative scale factor, the image is on the opposite side of the centre from the object. The size comes from the number without its sign. Scale factor -2 makes every length twice as long.

The working is the same as for a positive scale factor, with k negative.

Write it as an enlargement of scale factor -2. Do not write a negative enlargement of scale factor 2.

image point = centre + k × (object point - centre), k < 0

Examiner insight

“It is worth noting that ‘a negative enlargement of scale factor 2’ was a very common incorrect description which was given just one mark (for enlargement), rather than two marks for ‘an enlargement of scale factor –2’.” “There were fewer fully correct answers seen to part (b), with an incorrect scale factor of 2, rather than –2, being quite common.”

Worked example 13 marks

Do not use a calculator. The point P(3, 1) is enlarged with scale factor -2 and centre (1, 2). Write down the coordinates of the image of P.

  1. 1

    vector from the centre to P: (3 - 1, 1 - 2) = (2, -1) [1]

  2. 2

    multiply by -2: (-4, 2) [1]

  3. 3

    add to the centre: P' = (1 - 4, 2 + 2) = (-3, 4) [1]

7. Combinations of transformations

A combination applies one transformation, then a second to the image. Work in the order the question gives.

The order can change the final position. Write down the image after the first step before you start the second.

A combination is not a single transformation. If you are asked to describe one, give one.

Examiner insight

“Some did not know how to combine the transformations and it was common to see the original shape both reflected and translated.” “A few candidates did the translation the wrong way round.”

Worked example 13 marks

Do not use a calculator. The point P(1, 3) is rotated through 90° clockwise about the origin. The image is then reflected in the line y = 1. Find the final position of P.

  1. 1

    rotation first: (x,y) → (y, -x) gives (3, -1) [1]

  2. 2

    (3, -1) is 2 units below y = 1, so its image is 2 units above the line [1]

  3. 3

    final position = (3, 3) [1]

Recap

  • Describe fully: one transformation only, with every detail.
  • Translation: column vector, top number is x, bottom number is y.
  • Reflection: equation of the mirror line. Points are equally far from it on opposite sides.
  • Rotation: angle, direction and centre. 180° needs no direction.
  • Enlargement: centre and scale factor. A fractional scale factor makes the image smaller.
  • Extended: a negative scale factor puts the image on the opposite side of the centre.
  • Extended: in a combination, do the steps in the order given.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. State the three things you must give to describe a rotation fully.3 marks
  2. Do not use a calculator. The point P is at (2, -3). It is translated by the vector beginpmatrix -4 5 endpmatrix. Write down the coordinates of the image of P.2 marks
  3. Do not use a calculator. The point A is at (3, 5). It is reflected in the line x = 1. Write down the coordinates of the image of A.2 marks
  4. Do not use a calculator. The point R is at (4, 1). It is rotated through 90° clockwise about the origin. Write down the coordinates of the image of R.2 marks
  5. Triangle ABC has vertices A(1, 1), B(-3, 1) and C(-3, 2). It is rotated through 180° about the vertex A. Find the coordinates of the images of B and C.3 marks
  6. Do not use a calculator. The point T is at (5, 4). It is enlarged with scale factor 3 and centre (2, 1). Write down the coordinates of the image of T.3 marks
  7. Do not use a calculator. A triangle has a side of 12 cm. After an enlargement the matching side of the image is 3 cm. Write down the scale factor.2 marks
  8. Do not use a calculator. The point P is at (4, 3). It is enlarged with scale factor -2 and centre (1, 1). Write down the coordinates of the image of P.3 marks
  9. A lens model uses a grid. The object is triangle A with vertices (1, 2), (3, 2) and (1, 5). Its image is triangle B. The three vertices map, in the same order, onto (7, -1), (3, -1) and (7, -7). (a) Describe fully the single transformation that maps A onto B. (b) Describe fully the single transformation that maps B onto A.5 marks
  10. A machine prints a triangular tile T with vertices P(3, 1), Q(5, 1) and R(5, 2). It then moves the tile in two steps. Step 1 is a reflection in the line y = x. Step 2 is a reflection in the line x = 2. (a) Find the coordinates of the vertices of the tile after both steps. (b) Describe fully the single transformation that maps T onto its final position. (c) The machine is set wrongly and does Step 2 before Step 1. Find the final position of Q.7 marks

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